The blood vessels in your nose also expand to keep up with their job of warming up the air that you breathe in before it goes further down your respiratory tract. That in turn further activates your mucous producing cells, who among other things help moisten the air you breathe. All of this results in a runny nose!

Does it also have anything to do with the tendency of your eyes to water when you're out in the cold? I know other excess tears often end up running out the nose. (And if so, why do your eyes water when you're out in the cold?)

Sometimes eyes are watery because the normal moist layer protecting the eye is actually too dry. Normally, this layer is made of water and oil, which helps lubricate the eye and keeps it from drying out. When there is less oil available, the water begins to evaporate, and the tear glands secrete more to refresh. Ironically, you then get watery eyes as a result of having dry eyes. Windy days make many people's eyes water because this moist layer is drying out more quickly. Not sure if cold without wind would do the same, simply because it's dryer, but possibly?

I understand that, but my parents and I were having an argument that colds don't stem from being cold, but I couldn't refute the "fact" that people get colds from being cold. Is there any sort of relation? Like is the immune system weaker, or do colds spread faster in the cold?

Closer contact with more people indoors is the standard answer, but last month was published an interesting study on the transmission of the influenza virus. Turns out lower humidity winter indoor air (probably) permits the virus to survive longer in sneeze-mist. Obviously the flu and cold virions are built quite differently, but mode of transmission is similar, so both may benefit.

The reason for this phenomenon is threefold: One, as others have mentioned, people tend to stay indoors with other people more when it's cold out. Two, when humidity and temperature drop, viruses spread more easily, as shown by this study. Three, when you are cold, you are more susceptible to viral attack, as shown by the study discussed here.